Find the value of :

Options

A. `sin 74^(@)`
B. `cos 106^(@)`
C. `sin 15^(@)`
D. `cos 75^(@)`

Explanation

To solve the question, we need to find the values of the following trigonometric functions:

1. \( \sin 45^\circ \)
2. \( \sin 74^\circ \)
3. \( \sin 15^\circ \)
4. \( \cos 75^\circ \)

Let's go through each step one by one.

### Step 1: Find \( \sin 45^\circ \)
The value of \( \sin 45^\circ \) is a well-known trigonometric value:
\[ \sin 45^\circ = \frac{1}{\sqrt{2}} \quad \text{or} \quad \frac{\sqrt{2}}{2} \]

### Step 2: Find \( \sin 74^\circ \)
We can express \( \sin 74^\circ \) using the double angle formula:
\[ \sin 74^\circ = \sin(2 \times 37^\circ) = 2 \sin 37^\circ \cos 37^\circ \]
From trigonometric tables or calculations, we know:
\[ \sin 37^\circ = \frac{3}{5}, \quad \cos 37^\circ = \frac{4}{5} \]
Substituting these values:
\[ \sin 74^\circ = 2 \times \frac{3}{5} \times \frac{4}{5} = \frac{24}{25} \]

### Step 3: Find \( \sin 15^\circ \)
We can express \( \sin 15^\circ \) using the sine difference formula:
\[ \sin 15^\circ = \sin(45^\circ - 30^\circ) = \sin 45^\circ \cos 30^\circ - \cos 45^\circ \sin 30^\circ \]
Substituting known values:
\[ \sin 45^\circ = \frac{1}{\sqrt{2}}, \quad \cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \cos 45^\circ = \frac{1}{\sqrt{2}}, \quad \sin 30^\circ = \frac{1}{2} \]
Thus,
\[ \sin 15^\circ = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} - \frac{1}{\sqrt{2}} \cdot \frac{1}{2} = \frac{\sqrt{3} - 1}{2\sqrt{2}} \]

### Step 4: Find \( \cos 75^\circ \)
We can express \( \cos 75^\circ \) using the cosine addition formula:
\[ \cos 75^\circ = \cos(45^\circ + 30^\circ) = \cos 45^\circ \cos 30^\circ - \sin 45^\circ \sin 30^\circ \]
Substituting known values:
\[ \cos 75^\circ = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} - \frac{1}{\sqrt{2}} \cdot \frac{1}{2} = \frac{\sqrt{3} - 1}{2\sqrt{2}} \]

### Summary of Values
1. \( \sin 45^\circ = \frac{1}{\sqrt{2}} \)
2. \( \sin 74^\circ = \frac{24}{25} \)
3. \( \sin 15^\circ = \frac{\sqrt{3} - 1}{2\sqrt{2}} \)
4. \( \cos 75^\circ = \frac{\sqrt{3} - 1}{2\sqrt{2}} \)